Conical surface

In geometry, a conical surface is a three-dimensional surface formed from the union of lines that pass through a fixed point and a space curve.

An elliptic cone, a special case of a conical surface

Definitions

A (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex. Each of those lines is called a generatrix of the surface. The directrix is often taken as a plane curve, in a plane not containing the apex, but this is not a requirement.^[1]

In general, a conical surface consists of two congruent unbounded halves joined by the apex. Each half is called a nappe, and is the union of all the rays that start at the apex and pass through a point of some fixed space curve.^[2] Sometimes the term "conical surface" is used to mean just one nappe.^[3]

Special cases

If the directrix is a circle ${\displaystyle C}$, and the apex is located on the circle's axis (the line that contains the center of ${\displaystyle C}$ and is perpendicular to its plane), one obtains the right circular conical surface or double cone.^[2] More generally, when the directrix ${\displaystyle C}$ is an ellipse, or any conic section, and the apex is an arbitrary point not on the plane of ${\displaystyle C}$, one obtains an elliptic cone^[4] (also called a conical quadric or quadratic cone),^[5] which is a special case of a quadric surface.^[4][5]

Equations

A conical surface ${\displaystyle S}$ can be described parametrically as

${\displaystyle S(t,u)=v+uq(t)}$,

where ${\displaystyle v}$ is the apex and ${\displaystyle q}$ is the directrix.^[6]

Related surface

Conical surfaces are ruled surfaces, surfaces that have a straight line through each of their points.^[7] Patches of conical surfaces that avoid the apex are special cases of developable surfaces, surfaces that can be unfolded to a flat plane without stretching. When the directrix has the property that the angle it subtends from the apex is exactly ${\displaystyle 2\pi }$, then each nappe of the conical surface, including the apex, is a developable surface.^[8]

A cylindrical surface can be viewed as a limiting case of a conical surface whose apex is moved off to infinity in a particular direction. Indeed, in projective geometry a cylindrical surface is just a special case of a conical surface.^[9]

See also

• Conic section
• Quadric

References

1. Adler, Alphonse A. (1912), "1003. Conical surface", The Theory of Engineering Drawing, D. Van Nostrand, p. 166
2. Wells, Webster; Hart, Walter Wilson (1927), Modern Solid Geometry, Graded Course, Books 6-9, D. C. Heath, pp. 400–401
3. Shutts, George C. (1913), "640. Conical surface", Solid Geometry, Atkinson, Mentzer, p. 410
4. Young, J. R. (1838), Analytical Geometry, J. Souter, p. 227
5. Odehnal, Boris; Stachel, Hellmuth; Glaeser, Georg (2020), "Linear algebraic approach to quadrics", The Universe of Quadrics, Springer, pp. 91–118, doi:10.1007/978-3-662-61053-4_3, ISBN 9783662610534
6. Gray, Alfred (1997), "19.2 Flat ruled surfaces", Modern Differential Geometry of Curves and Surfaces with Mathematica (2nd ed.), CRC Press, pp. 439–441, ISBN 9780849371646
7. Mathematical Society of Japan (1993), Ito, Kiyosi (ed.), Encyclopedic Dictionary of Mathematics, Vol. I: A–N (2nd ed.), MIT Press, p. 419
8. Audoly, Basile; Pomeau, Yves (2010), Elasticity and Geometry: From Hair Curls to the Non-linear Response of Shells, Oxford University Press, pp. 326–327, ISBN 9780198506256
9. Giesecke, F. E.; Mitchell, A. (1916), Descriptive Geometry, Von Boeckmann–Jones Company, p. 66